This is hard to explain in a quick reply, because there are so many rules and variations of each game, but in a real quick nutshell:
Trap:
Shooters stand behind a trap house from 16 yards mostly. The birds are thrown away from you at a set max. height and speed. The birds will go straight, right and left in any random order. Up to 5 shooters stand side by side at a seperate station. Each shooter will call for a bird starting with station 1. After each shooter shoots 5 birds, you rotate to the next station and so on... at the end you will have fired 25 shots. If you broke 22 out of 25 you will move back to the 22 yard line and shoot a second round. This is called shooting your handicap. You shoot 25 from 16 yards, then your handicap, then 16 yards, then another handicap. Total score - how many you broke out of 100. There is also doubles trap, and variations of number of birds shot and several games that relate to trap.
Skeet is shot kinda similar only the traps are set in 2 towers. One on the right of you, one on your left. In a nutshell...the birds are launched left to right and right to left at different levels. I don't shoot skeet...so someone will help me out later I'm sure. Thanks guys!
Sporting clays is my game... it is kinda like golf with a gun. Different stations are set up and each shooter will travel through, stopping to shoot each station. Birds are launched from all angles and all directions. I've seen just about everything you can imagine. Different size birds are launched, rabbits, different colors, different speeds...it can get kinda crazy. But it's a ball to shoot.
All of the shotgun sports require skill. IMHO trap is a thinking mans game. The birds are not that hard to hit, but you will mess up and lose to someone that keeps a clear head. From what I hear, skeet is more of a speed sport than trap. To me sporting clays is the most fun, and is a instinctive reaction type of shooting that is kinda like bird or rabbit hunting.
As I said, all three are fun..and there are many different games that spin off of those 3 games. This was just a quick reply...more folks will chime in and talk about their favorite in more detail I'm sure.